Variations sur un thème de Aldama et Shelah
Résumé
We consider a group G that does not have the independence property and study the definability of certain subgroups of G
using parameters from a fixed elementary extention G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup.
We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived
length ℓ. The subgroup S is also contained in an externally definable subgroup X∩G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that
does not have the independence property.
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